Civil Engineering Reference
In-Depth Information
Probably the most descriptive way to contrast the two methods is to note that
the finite difference method models the differential equation(s) of the problem
and uses numerical integration to obtain the solution at discrete points. The finite
element method models the entire domain of the problem and uses known phys-
ical principles to develop algebraic equations describing the approximate solu-
tions. Thus, the finite difference method models differential equations while the
finite element method can be said to more closely model the physical problem at
hand. As will be observed in the remainder of this text, there are cases in which
a combination of finite element and finite difference methods is very useful and
efficient in obtaining solutions to engineering problems, particularly where dy-
namic (transient) effects are important.
1.3 A GENERAL PROCEDURE FOR FINITE
ELEMENT ANALYSIS
Certain steps in formulating a finite element analysis of a physical problem are
common to all such analyses, whether structural, heat transfer, fluid flow, or
some other problem. These steps are embodied in commercial finite element
software packages (some are mentioned in the following paragraphs) and are
implicitly incorporated in this text, although we do not necessarily refer to the
steps explicitly in the following chapters. The steps are described as follows.
1.3.1 Preprocessing
The preprocessing step is, quite generally, described as defining the model and
includes
Define the geometric domain of the problem.
Define the element type(s) to be used (Chapter 6).
Define the material properties of the elements.
Define the geometric properties of the elements (length, area, and the like).
Define the element connectivities (mesh the model).
Define the physical constraints (boundary conditions).
Define the loadings.
The preprocessing (model definition) step is critical. In no case is there a better
example of the computer-related axiom “garbage in, garbage out.” A perfectly
computed finite element solution is of absolutely no value if it corresponds to the
wrong problem.
1.3.2 Solution
During the solution phase, finite element software assembles the governing alge-
braic equations in matrix form and computes the unknown values of the primary
field variable(s). The computed values are then used by back substitution to
